Let $\Omega\subset \mathbb{R}^N$ with a smooth boundary $\partial\Omega$ and consider the elliptic PDE \begin{alignat*}{4} \mbox{D.E.} & \quad{\displaystyle \Delta u -u =0 },&& \quad x\in\Omega\setminus\partial\Omega, \nonumber\\ \mbox{B.C.}& \quad{\displaystyle \frac{\partial u}{\partial \eta}+\kappa(x)u=0},&& \quad x\in\partial\Omega, \end{alignat*} where it is assumed that function $\kappa(x)\in L^p(\partial\Omega)$ is strictly positive on $\partial\Omega$ and is bounded by a positive constant $M$. Is it correct to assume that a solution to this linear problem exists in weak form?
Does the weak form representation in \begin{equation*} -\int_{\Omega}\nabla u \cdot\nabla\phi\,{\mbox d}x -\int_{\partial\Omega}\kappa(x)u\phi\,{\mbox d}s + \int_\Omega u\phi\,{\mbox d}x =0 \quad \forall \phi\in H^1. \end{equation*} make sense?
Am I assuming that the test function belongs to the correct space?
Is it possible to obtain a priori estimate for this solution to show that it is bounded?
I am aware that these questions are well established in the case where $\kappa$ is a constant but I am struggling to find references for the case where $\kappa$ depends on $x$ and is not necessarily continuous.